1. REPRESENTING & INTERPRETING DATA C A S I O | P U T VA L U E B A C K I N T H E E Q U A T I O N 2. PATTERNS AND FUNCTIONS Investigation 7.2:ȱ ǰȱǰȱdz 3. PROPORTIONAL REASONING ȱĜȱȱȱǻǼȱȱ¢ȱȱȱ ȱǯȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱȱ¢ȱȱ ȱȱȱ ȱȱŖǯŞřŖǯȱȱȱ ǰȱȱ ȱȱŞřȱȱȱȱȂȱ¢ȱȱȱ ȱȱȱǯ Reference: http://golf.about.com/od/faqs/f/cor.htm 4. LINEAR FUNCTIONS The coefficient of restitution (COR) indicates the bounciness of an object. If an object has a COR of 1, when it is dropped from a certain height, its return bounce reaches the original height. The closer the COR is to 1, the higher it bounces back. The closer the COR is to 0, the less it bounces back. 5. SYSTEMS OF LINEAR EQUATIONS What do you think? Can you rank these from highest to lowest COR? Explain your reasoning. ȱ ȱ ȱȱ ȱ ȱȱ ȱȱ ȱ ȱȱȱ ȱ ȱ ȱȱ ȱ ȱ ȱ 6. LINEAR PROGRAMMING For this investigation, choose a ball that you believe has a relatively high COR. Drop it onto a hard, level floor from a height of 2 meters (there’s nothing magical about 2 meters; we have chosen it arbitrarily) and explore the relationship between the height of the bounce and how many times it has bounced. Round each value to the nearest centimeter. 7. EXPONENTIAL FUNCTIONS Aȱ ȱ ȱ¢ȱ ȱȱ¢ȱȱȱȱ ȱȱ¢ȱ ȱ¢ȱȱǯ Bȱ ȱȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱȱǯȱȱȱ ȱȱȱȱȱ¡ȱȱ ȂȱĜȱȱǯȱ Cȱ ȱȱĴȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱǯȱȱȱǯ 8. QUADRATIC FUNCTIONS ŘŝŖ 1. REPRESENTING & INTERPRETING DATA C A S I O | W W W. C A S I O E D U C A T I O N . C O M BOUNCE, BOUNCE, BOUNCE… (CONTINUED) Fȱ ȱȱȱȱȱȱȱȱȱŗŖȱǰȱȱȱśŖȱ ǯȱȱȱȱȱȱȱȱȱǵȱ¢ȱȱ ¢ȱǵ H ȱ ȱ¢ȱȱȱȱ¢ȱȱȱ ȱŗŖŖȱǵȱ ¡ǯ 4. LINEAR FUNCTIONS Gȱ ȱȱȱȱȱ ȱ¢ȱȱȱ ȱȱȱȱȱȱ ȱ¡ȱȱȱśȱȱȱŖǯŗȱǯȱ ȱ ȱ¢ȱȱęȱ ȱǯ 3. PROPORTIONAL REASONING Eȱ ȱȱȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱǯȱȱȱȱȬȱȱȱȱ ¡ȱǵȱȱȱȱȱȱǵȱȱȱȱȬ ȱ ȱȱȱȱȱǵ 2. PATTERNS AND FUNCTIONS Dȱ ȱȱȱȱȱȱȱȱȱȱ¡ȱ ǵȱȱ¡ȱ¢ȱǯ Iȱ ¡ȱȱȱȱȱ¢ȱȱěȱǯ 5. SYSTEMS OF LINEAR EQUATIONS Jȱ ȱĜȱȱȱȱęȱȱȱȱȱȱȱȱ ȱȱȱǰȱhǰȱȱȱȱǰȱHǰȱCOR == Hh ǯȱ¡ȱ ȱĜȱȱȱȱȱȱ¢ȱȱȱȱǯȱȱ 6. LINEAR PROGRAMMING 7. EXPONENTIAL FUNCTIONS 8. QUADRATIC FUNCTIONS Řŝŗ 1. REPRESENTING & INTERPRETING DATA C A S I O | P U T VA L U E B A C K I N T H E E Q U A T I O N 2. PATTERNS AND FUNCTIONS Table for Investigation 7.2:ȱ ǰȱǰȱȱdz Complete the following table to investigate the height of the bounce as it relates to the number of times the ball bounced. 3. PROPORTIONAL REASONING BOUNCE NUMBER BOUNCE HEIGHT (CENTIMETERS) RATIO (BOUNCE HEIGHT DIVIDED BY PREVIOUS BOUNCE HEIGHT) 0 ---- 4. LINEAR FUNCTIONS 1 2 5. SYSTEMS OF LINEAR EQUATIONS 3 6. LINEAR PROGRAMMING 4 5 7. EXPONENTIAL FUNCTIONS 6 8. QUADRATIC FUNCTIONS ŘŝŘ 1. REPRESENTING & INTERPRETING DATA C A S I O | W W W. C A S I O E D U C A T I O N . C O M Aȱ ȱ ȱ¢ȱ ȱȱ¢ȱȱȱȱ ȱȱ¢ȱ ȱ¢ȱȱǯ BOUNCE HEIGHT (CM) RATIO 0 200 ---- 1 150 0.75 2 114 0.76 3 92 0.80 4 70 0.76 5 57 0.81 6 45 0.79 6. LINEAR PROGRAMMING BOUNCE NUMBER 5. SYSTEMS OF LINEAR EQUATIONS The data in the Bounce Height column is from a group of students who did this investigation with a “bouncy ball,” which has a relatively high COR. To determine the ratio, we simply used the Run menu and divided the Bounce Height by the Bounce Height above it, rounding to the nearest hundredth. 4. LINEAR FUNCTIONS Bȱ ȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱǯȱȱȱ ȱȱȱȱȱȬ ¡ȱȱȂȱĜȱȱǯȱ 3. PROPORTIONAL REASONING There are different ways to collect the data. We have found that students have the best success when they try to determine the highest point after a bounce and then catch the ball, typically trying three times from a given height. For example, if the first bounce reaches 150 cm, for the second bounce, they will release the ball from 150 cm and try to determine how high it bounces back. If you have a motion detector or digital camera, you may be able to be more accurate. 2. PATTERNS AND FUNCTIONS Sample Solution:ȱ ǰȱǰȱȱdz 7. EXPONENTIAL FUNCTIONS 8. QUADRATIC FUNCTIONS Řŝř 1. REPRESENTING & INTERPRETING DATA C A S I O | P U T VA L U E B A C K I N T H E E Q U A T I O N BOUNCE, BOUNCE, BOUNCE… (CONTINUED) 2. PATTERNS AND FUNCTIONS Cȱ ȱȱĴȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱǯȱȱȱǯ 3. PROPORTIONAL REASONING To create the scatterplot, we went to the Statistics menu and entered our data into two lists, one for the Bounce Number and the other for the Bounce Height. From there, if B(GRPH) is not an option, then press 5 until you have backed out from wherever you were, and then press F, if needed, so that B(GRPH) is available. Then: Press B(GRPH) and F(SET) to set the graph up. For StatGraph1, set up a Scatterlot, using the list for the Bounce Number as your XList:, the list for your Bounce Height for your YList:, and set the Frequency at 1. 4. LINEAR FUNCTIONS Press 5 to leave the SET screen and B(GPH1) to draw the graph. 5. SYSTEMS OF LINEAR EQUATIONS The plot shows a descending curve whose slope is less and less steep as we view the graph from left to right. The y-intercept is the first point on the graph. 6. LINEAR PROGRAMMING Dȱ ȱȱȱȱȱȱȱȱȱȱ¡ȱ ǵȱ¡ȱ¢ȱǯ 7. EXPONENTIAL FUNCTIONS The bounciness can indeed be modeled by an exponential function. We expect the ball to bounce back a certain percent of its maximum height in each succeeding bounce. For example, if a ball bounces back 60% of its maximum height and we began at 200 cm, then on the first bounce we would expect it to reach back to 60% of 200 cm, on its second bounce we expect 60% of (60% of 200), and we continue to take 60% of the preceding value. This fits the general form for an exponential function of y = abx, where x represents the bounce number and y represents the maximum height of the bounce. In this hypothetical example, we would have y = 200 • 0.60x. 8. QUADRATIC FUNCTIONS ŘŝŚ 1. REPRESENTING & INTERPRETING DATA C A S I O | W W W. C A S I O E D U C A T I O N . C O M BOUNCE, BOUNCE, BOUNCE… (CONTINUED) Press B(CALC)F 4. LINEAR FUNCTIONS Another way to find an algebraic function is to allow the calculator to find the best-fit exponential regression model. While looking at the scatterplot: 3. PROPORTIONAL REASONING We will tackle this in two different ways. The first way, which may help students develop a more intuitive feel for a decreasing exponential function, we recognize that our data contains (0, 200). We then average the six ratios in our third column to determine the mean percent of the height the ball bounces back. Finding this average gives us approximately 78%. Combining these two ideas, we obtain an algebraic model of y = 200 • 0.78x. Here our b value is 0.78, which gives us the percent (if we expressed it as 78%) of the height the ball returns to, on average. Our value for a is 200, the y-intercept, showing that after 0 bounces, the ball reached 200 cm. 2. PATTERNS AND FUNCTIONS Eȱ ȱȱȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱǯȱȱȱȱȬȱȱȱȱ ¡ȱǵȱȱȱȱȱȱǵȱȱȱȱȬ ȱ ȱȱȱȱȱǵ 6(EXP)H(abx) to pick the form of the exponential model we wish to use. 5. SYSTEMS OF LINEAR EQUATIONS 7. EXPONENTIAL FUNCTIONS Fȱ ȱȱȱȱȱȱȱȱȱŗŖȱǰȱȱȱśŖȱ ǯȱȱȱȱǰȱȱȱȱȱǵȱ¢ȱȱ ¢ȱǵ 6. LINEAR PROGRAMMING This gives us the algebraic equation of approximately y = 193 • 0.78x. Though at first this may seem less accurate than our previous model because the y-intercept is no longer 200, we do note that we are not that far off and know that the calculator has taken each data point into better account than we were able to before. Note, that we again find that the ball, on average, bounces back about 78% of its previous maximum height. We will take advantage of the calculator to address this. While looking at the screen above in which the regression model was determined: 8. QUADRATIC FUNCTIONS Řŝś 1. REPRESENTING & INTERPRETING DATA C A S I O | P U T VA L U E B A C K I N T H E E Q U A T I O N BOUNCE, BOUNCE, BOUNCE… (CONTINUED) 2. PATTERNS AND FUNCTIONS Press J(COPY) to copy the model into the GRAPH and TABLE menus. Press = to paste the function, being sure not to overwrite any function in the calculator you wish to keep. You will be returned to the previous screen, but the function will now be stored in your calculator. 3. PROPORTIONAL REASONING Press A for the Main Menu followed by for the Table Menu. Highlight the equal sign of your function by pressing B(SELECT). You can SET UP the table if you desire, but that actually isn’t necessary; simply press F(TABLE) to view the table. 4. LINEAR FUNCTIONS With the cursor on any value in the X column, you can type = then move to another x-value and type =. At 10 bounces, we expect a maximum height of 16.4 cm, and at 50 bounces, the ball should get back to 0.00086 cm. 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING Though if we went far, far down the table, we would reach a height of 0 cm, this is because of rounding error. According to our model, the ball will always bounce back some, because taking a part (78%) of a positive number will always give a positive number. However in the real world, the ball will stop bouncing because of the energy lost to heat and friction. The model is not perfect, but it still provides an excellent tool for describing the world. 7. EXPONENTIAL FUNCTIONS Gȱ ȱȱȱȱȱ ȱ¢ȱȱȱ ȱȱȱȱ ȱȱȱ¡ȱȱȱśȱȱȱŖǯŗȱǯȱ ȱ ȱ¢ȱȱ ȱęȱȱǯ The X-CAL function on the calculator is an amazing tool. From the GRAPH menu, make sure the desired function is selected. Then: 8. QUADRATIC FUNCTIONS ŘŝŜ 1. REPRESENTING & INTERPRETING DATA C A S I O | W W W. C A S I O E D U C A T I O N . C O M BOUNCE, BOUNCE, BOUNCE… (CONTINUED) Type in =. If you get the message “not found”, your window needs adjusting and press 6(V-Window) to adjust the view window. Change the Xmax: to 50, pressing = after you type in the value. Press 5 to back out of the V-Window and F to redraw the graph. that the calculator tells you that the x-values associated with 5 cm and 0.1 cm are 14.8, and 30.7 respectively. This means on the 15th and 31st bounces, the ball will no longer reach 5 cm and 0.1 cm, respectively. 3. PROPORTIONAL REASONING Repeat the steps above to get to X-CAL and try again. You should find 2. PATTERNS AND FUNCTIONS Press F(DRAW)J(G-SOLV), F, and H(X-CAL). 4. LINEAR FUNCTIONS The b-value for our exponential model would remain the same, but the a value should become 100 instead of 200. Again, the a value represents the y-intercept, the initial height from which the ball is released. 5. SYSTEMS OF LINEAR EQUATIONS H ȱ ȱ¢ȱȱȱȱ¢ȱȱȱ ȱŗŖŖȱǵȱ ¡ǯ Iȱ ¡ȱȱȱȱȱ¢ȱȱěȱǯ Performing the arithmetic on the calculator, we find the COR for the bouncy ball we used to be approximately 0.88. In regard to the ranking requested at the beginning of this problem, students should probably have been able to put the balls into a reasonable order. 8. QUADRATIC FUNCTIONS Řŝŝ 7. EXPONENTIAL FUNCTIONS Jȱ ȱĜȱȱȱȱęȱȱȱȱȱȱȱȱȱ h = ȱȱǰȱhǰȱȱȱȱǰȱHǰȱCOR =ȱ ǯȱ¡ȱȱ H Ĝȱȱȱȱȱȱ¢ȱȱȱȱǯ 6. LINEAR PROGRAMMING Our data were taken with a very bouncy ball. With others, the value for a should remain close to 200, but the b-value will likely be significantly less. For example, with a baseball, you probably could not collect data for six bounces because the maximum height would appear to be 0 fairly quickly. INTRODUCTION DATA ENTRY C A S I O | W W W. C A S I O E D U C A T I O N . C O M Bǯȱ ȱȱȱǰȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱ¢ȱȱ¢ȱěǯȱȱ¢ȱ ęȱȱȱȱȱěǰȱȱȱȱęȱ ȱȱ ěȱǯ 6. DIFFERENCES BETWEEN 2 CROPS Cǯȱ ȱȱȱǰȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱȱ¢ȱȱ¢ȱěȬ ǯȱȱ¢ȱęȱȱȱȱȱěǰȱȱȱȱęȱ ȱȱěȱǯȱȱ 5. REGRESSION MODELS AND ANALYSIS Aǯȱ ȱ¢ȱ¡ȱȱȱȱȱěȱȱȱȱȱȱȬ ȱȱȱȱȱȱȱȱȱěȱ¢ǵȱ ȱȱ ¢ȱȱǵȱ¡ǯ 4. UNIVARIATE INFERENCES Are you concerned about what you eat? Many people are, though many people are not. Hotdogs are quite popular, particularly at picnics and sporting events. Are all hotdogs created equally? For this investigation, we look at 54 major hotdog brands coming in three major types: beef, poultry, or a more generic meat (mostly pork and beef, but as much as 15% poultry). More specifically, in this investigation we’ll take a look at the mean number of calories and the mean amount of sodium (measured in milligrams) per hotdog in the three types. 3. NORMAL DISTRIBUTION Reference: http://www.nytimes.com/2010/07/13/sports/baseball/13hotdogs.html 2. COUNTING AND THE BINOMIAL DISTRIBUTION ȱȱȱȱȱȱȱȱȱȱ ȱȱȱǯȱȱ¡ȱȱ ȱȱŘȱȱ ȱȱȱȱȱ¢ǰȱȱŖǯŜŚȱȱȱǯȱ ȱȱĴǰȱȱȱȱȱȱ Ȭȱȱȱȱȱǰȱ ȱŖǯśŝȱ ȱȱȱȱǰȱ ȱ¢ȱȱȱ ȱǰȱȱȱȱȱ ȱǯ 1. DESCRIPTIVE TECHNIQUES Investigation 7.1:ȱ 7. ANOVA AND CHI-SQUARE 8. EXTENDING STATISTICAL THINKING ŗşŝ INTRODUCTION DATA ENTRY C A S I O | P U T VA L U E B A C K I N T H E E Q U A T I O N 1. DESCRIPTIVE TECHNIQUES HOT DOG BEEF Cal 186 181 176 149 184 190 158 139 175 148 152 111 141 153 190 157 131 149 135 132 Sod 495 477 425 322 482 587 370 322 479 375 330 300 386 401 645 440 317 319 298 253 2. COUNTING AND THE BINOMIAL DISTRIBUTION MEAT Cal 173 191 182 190 172 147 146 139 175 136 179 153 107 195 135 140 138 Sod 458 506 473 545 496 360 387 386 507 393 405 372 144 511 405 428 339 POULTRY 3. NORMAL DISTRIBUTION Cal 129 132 102 106 94 102 87 99 107 113 135 142 86 143 152 146 144 Sod 430 375 396 383 387 542 359 357 528 513 426 513 358 581 588 522 545 Reference: http://lib.stat.cmu.edu/DASL/Datafiles/Hotdogs.html, which has taken the information from Moore and McCabe’s Introduction to the Practice of Statistics (1989). The original source for the data is: Consumer Reports, June 1986, pp. 366-367. 4. UNIVARIATE INFERENCES 5. REGRESSION MODELS AND ANALYSIS 6. DIFFERENCES BETWEEN 2 CROPS 7. ANOVA AND CHI-SQUARE 8. EXTENDING STATISTICAL THINKING ŗşŞ INTRODUCTION DATA ENTRY C A S I O | W W W. C A S I O E D U C A T I O N . C O M Aǯȱ ȱ¢ȱ¡ȱȱȱȱȱěȱȱȱȱȱȱ ȱȱȱȱȱȱȱȱȱěȱ¢ǵȱ ȱ ȱ¢ȱȱǵȱ¡ǯ 5. REGRESSION MODELS AND ANALYSIS To enter the data, we need to use three lists. We will use List 1 for the type (1 = beef, 2 = meat, and 3 = poultry), List 2 for the calories, and List 3 for the sodium. The screens below show the beginning and the end of the data. 4. UNIVARIATE INFERENCES Though we could run three t-tests, comparing beef with meat, beef with poultry, and meat with poultry, the PRIZM allows us to conduct an ANOVA (Analysis of Variance). This test is useful when trying to determine if there is a statistically significant difference in means when there are more than two groups. Not only is one test more efficient than conducting a series of t-tests, but when running multiple t-tests, there is a much greater chance of a Type I error. For example, if we run three t-tests each with an alpha value set at 0.05, the Bonferroni Inequality tells us the probability of a Type I error may be as high as 0.15. When we have several groups, this can lead to unacceptable probabilities of forming an incorrect conclusion. 3. NORMAL DISTRIBUTION Bǯȱ ȱȱȱǰȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱ¢ȱȱ¢ȱěǯȱȱ¢ȱ ęȱȱȱȱȱěǰȱȱȱȱęȱ ȱȱ ěȱǯ 2. COUNTING AND THE BINOMIAL DISTRIBUTION Answers will vary. Some may expect the generic “meat” to have the highest number of calories and the poultry the fewest, but many may have no idea. We confess that we had no prediction ahead of time for the mean amount of sodium in the three types. 1. DESCRIPTIVE TECHNIQUES One Solution:ȱ 6. DIFFERENCES BETWEEN 2 CROPS 8. EXTENDING STATISTICAL THINKING ŗşş 7. ANOVA AND CHI-SQUARE The null hypothesis for the ANOVA is that all group means are equal. Before we run the test, however, we need to determine our alpha value. We will, arbitrarily, use a value of 0.05. If p is less than that, then we will reject the null hypothesis and conclude that at least one of the means is different from the others. If p is 0.05 or larger, then we will not have sufficient evidence to conclude that the means are different, and no further analysis will be necessary. To run the test, from the screen above right, INTRODUCTION DATA ENTRY C A S I O | P U T VA L U E B A C K I N T H E E Q U A T I O N 1. DESCRIPTIVE TECHNIQUES HOT DOG Press F twice for more options, 6(TEST)J(ANOVA). There is one factor (independent variable), the type of hotdog, even 2. COUNTING AND THE BINOMIAL DISTRIBUTION though there are three levels of that factor. The Factor is in List 1 and the Dependent variable, the calorie content, is in List 2. See below for the setup for the ANOVA. 3. NORMAL DISTRIBUTION Scroll down to Execute; press B(CALC). 4. UNIVARIATE INFERENCES 5. REGRESSION MODELS AND ANALYSIS 6. DIFFERENCES BETWEEN 2 CROPS For the independent variable (the type of hotdog, identified by A), there were three levels, and thus two degrees of freedom. The calculator also reports the sum of squares for this variable (the sum of squared deviations from the grand mean number of calories), and the mean sum of squares, which is the sum of squares divided by the degrees of freedom. These are sometimes referred to as our “between” statistics. The error refers to the “within” statistics. In general, ANOVA compares the differences between the groups to the differences within the groups. 7. ANOVA AND CHI-SQUARE Our main is focus is on the p-value, approximately 0.00000386, which is associated with the high F-ratio and the degrees of freedom shown. This, obviously, is quite small and clearly less than our preset alpha value. We therefore reject the null hypothesis and conclude that the mean values of the calorie content for the three types of hotdogs are not the same. We do not yet know, however, where the difference lies. 8. EXTENDING STATISTICAL THINKING ŘŖŖ INTRODUCTION DATA ENTRY C A S I O | W W W. C A S I O E D U C A T I O N . C O M To enter the data, we will use a sequence to pull the first 20 values in List 2 into List 4, the next 17 values in List 2 into List 5, and the final 17 values in List 2 into List 6. From the home STATISTICS screen, highlight List 4. Then, The entire command for the sequence is Seq(List 2[x], x, 1, 20, 1). List Highlight List 5 and use a similar process, though keep in mind that we want the 21st through the 37th values from List 2. Thus the command is Seq(List 2[x], x, 21, 37, 1). 5. REGRESSION MODELS AND ANALYSIS 2[x] identifies specific entries in List 2, with x as the variable. We will have x go from 1 (the first value) to 20 (the 20th value), counting by 1. Recall that "(List) can be used to generate “List.” Complete the sequence commands as shown below left, though note that the very last part of the command is not visible. Press = to complete the List; you may also wish to label the list. 4. UNIVARIATE INFERENCES Press :B(LIST)J(Seq). 3. NORMAL DISTRIBUTION In order to perform these t-tests, we need to arrange our data differently in the calculator. We will use List 4 for the calories of the beef hotdogs, List 5 for the meat hotdogs, and List 6 for the poultry. We know there were 20 brands of beef, 17 brands of meat, and 17 brands of poultry. 2. COUNTING AND THE BINOMIAL DISTRIBUTION There are many different tests that can be used as post-hoc efforts to a significant ANOVA. One possibility is to conduct a series of t-tests. If we are to do so, however, we should be aware of how many tests are warranted and adjust our alpha accordingly. Here there are three possible t-tests (beef vs. meat, beef vs. poultry, and meat vs. poultry), so we will divide our original alpha of 0.05 by 3. This is based on the idea of the Bonferroni Inequality discussed earlier. Thus for our post-hoc t-tests, we will set alpha at 0.0167. 1. DESCRIPTIVE TECHNIQUES HOT DOG 6. DIFFERENCES BETWEEN 2 CROPS 7. ANOVA AND CHI-SQUARE 8. EXTENDING STATISTICAL THINKING ŘŖŗ INTRODUCTION DATA ENTRY C A S I O | P U T VA L U E B A C K I N T H E E Q U A T I O N 1. DESCRIPTIVE TECHNIQUES HOT DOG Use a similar process to populate List 6 with the 38th through 54th values from List 2. Below are the beginning and end of the three lists. ••••• 2. COUNTING AND THE BINOMIAL DISTRIBUTION 3. NORMAL DISTRIBUTION We are now ready to conduct our t-tests to see where the differences lie. From the screen shown above, Press "5(QUIT) to return to the home STATISTICS screen. Press 6(TEST)H(t)H(2-SAMPLE). We’ll first test Beef against Meat, using a 2-tailed test. See part of the 4. UNIVARIATE INFERENCES setup below left. Using Stevens’ argument about pooling the variances if the group sizes are reasonably close, we have turned on the Pooled option. Scroll down to Execute; press B(CALC). ••••• 5. REGRESSION MODELS AND ANALYSIS To repeat this process to conduct the other tests, 6. DIFFERENCES BETWEEN 2 CROPS Press 5. Change the List(s) as appropriate to run the desired test. Then scroll down to Execute; press B(CALC). The screens below show the setup and results for Beef vs. Poultry. ••••• 7. ANOVA AND CHI-SQUARE 8. EXTENDING STATISTICAL THINKING ŘŖŘ INTRODUCTION DATA ENTRY C A S I O | W W W. C A S I O E D U C A T I O N . C O M Below are the setup and results for Meat vs. Poultry. 1. DESCRIPTIVE TECHNIQUES HOT DOG 2. COUNTING AND THE BINOMIAL DISTRIBUTION 8. EXTENDING STATISTICAL THINKING ŘŖř 7. ANOVA AND CHI-SQUARE We will again conduct an ANOVA, using an alpha value of 0.05. If the p-value is less than alpha, then we will reject the null hypothesis that the mean sodium content for the three groups is the same. If the p-value is 0.05 or greater, then we will fail to reject the null; we will not have sufficient evidence to conclude that the mean sodium content for the three groups is different. 6. DIFFERENCES BETWEEN 2 CROPS Cǯȱ ȱȱȱǰȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱȱ¢ȱȱ¢ȱěȬ ǯȱȱ¢ȱęȱȱȱȱȱěǰȱȱȱȱęȱ ȱȱěȱǯȱ 5. REGRESSION MODELS AND ANALYSIS Because a two-tailed test, as was done here, is more difficult to obtain significance with than a one-tailed test, we can go a little further here. We can state that the mean number of calories of a beef hotdog, at approximately 157 calories, is significantly more than the mean number of calories of a poultry hotdog, at approximately 119 calories. Similarly, the mean number of calories of a meat hotdog, at approximately 159 calories, is significantly more than the mean number of calories of a poultry hotdog. Thus, if you’re interested in lowering your intake of calories, you should select poultry hotdogs over beef or meat hotdogs. 4. UNIVARIATE INFERENCES However, for the other two tests, the p-value is well below our alpha of 0.0167. Consequently we can say that there is, on average, a significant difference in the mean number of calories between a beef hotdog and a poultry hotdog and between a meat hotdog and a poultry hotdog. 3. NORMAL DISTRIBUTION We see that the results comparing Lists 4 and 5 (Beef vs. Meat) are not significant, with a p-value of approximately 0.815. Consequently we cannot claim that, on average, the number of calories in a beef hotdog is different from the number of calories in a meat hotdog. INTRODUCTION DATA ENTRY C A S I O | P U T VA L U E B A C K I N T H E E Q U A T I O N 1. DESCRIPTIVE TECHNIQUES HOT DOG From the home Statistics screen, Press 6(TEST)J(ANOVA). The setup is similar to the one for calories, except the Dependent variable is now in List 3. 2. COUNTING AND THE BINOMIAL DISTRIBUTION 3. NORMAL DISTRIBUTION Scroll down to Execute; press B(CALC). ••••• 4. UNIVARIATE INFERENCES 5. REGRESSION MODELS AND ANALYSIS Our p-value, approximately 0.179, which is associated with the relatively small F-ratio and based on the degrees of freedom, is not less than alpha (0.05). We do not have evidence that the mean sodium content differs for beef, meat, and poultry hotdogs, so no further analysis is warranted. 6. DIFFERENCES BETWEEN 2 CROPS 7. ANOVA AND CHI-SQUARE 8. EXTENDING STATISTICAL THINKING ŘŖŚ
© Copyright 2025 Paperzz